Question

What is the best approximation of the solution to the system to the nearest integer values?

Responses

(7, −2)
begin ordered pair 7 comma negative 2 end ordered pair

(6, −2)
begin ordered pair 6 comma negative 2 end ordered pair

(−2, 6)
begin ordered pair negative 2 comma 6 end ordered pair

(−2, 7)
begin ordered pair negative 2 comma 7 end ordered pair
Graph of a system of linear equations. Equation 1 is 3 x minus 4 y equals negative 32. Equation 2 is 3 x plus 5 y equals 24.

Answer 1

To find the solution to the system of linear equations given by:

1. \( 3x - 4y = -32 \)
2. \( 3x + 5y = 24 \)

we can solve these equations simultaneously.

First, let's solve one of the equations for \(x\). We'll use the first equation for this:

\[
3x - 4y = -32
\]

Rearranging this gives:

\[
3x = 4y - 32 \implies x = \frac{4y - 32}{3}
\]

Now, we'll substitute \(x\) in the second equation:

\[
3\left(\frac{4y - 32}{3}\right) + 5y = 24
\]

This simplifies to:

\[
4y - 32 + 5y = 24
\]
\[
9y - 32 = 24
\]
\[
9y = 56
\]
\[
y = \frac{56}{9} \approx 6.222
\]

Next, substitute \(y\) back into the equation for \(x\):

\[
x = \frac{4\left(\frac{56}{9}\right) - 32}{3}
\]
\[
= \frac{\frac{224}{9} - 32}{3} = \frac{\frac{224}{9} - \frac{288}{9}}{3} = \frac{-\frac{64}{9}}{3} = -\frac{64}{27} \approx -2.370
\]

Therefore, the approximate solution to the system is:

\[
(x, y) \approx (-2.370, 6.222)
\]

Now, we will round both values to the nearest integers:

\[
x \approx -2 \quad \text{and} \quad y \approx 6
\]

Thus, the best approximation of the solution to the nearest integer values is:

\[
(-2, 6)
\]

So, the correct answer is:

\(\text{(−2, 6)}\)